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# Lecture14 - Todays Schedule Sample size and power for...

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20 June 2003 Biostatistics 6650--L14 1 Today’s Schedule Sample size and power for hypothesis testing Power, general 1-sample tests Power Sample size 2-sample tests Power Sample size References/Software Examples using Beyer and Cochran SS tables (handout) .

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20 June 2003 Biostatistics 6650--L14 2 Power You are planning an experiment and you want to give yourself the best possible chance of determining the truth. Correct decisions: reject Ho when we should---power issue FTR Ho when we should--0 issue Planning stage: what effects do you think are possible? what is a clinically meaningful effect ? What result do you need to proceed to the next stage? What result do you need to recommend a change in clinical practice? what sample size is required to make it all work?
20 June 2003 Biostatistics 6650--L14 3 Power You have just completed an experiment and did not reject Ho. How comfortable are you with the conclusion? Was the estimated effect near zero? Was the 95% CI for the effect size small enough to rule out all effects of interest? Was the negative result based on an n of 5,50 or 500? What power did you have to detect meaningful effects?

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20 June 2003 Biostatistics 6650--L14 4 Power Power=probability of rejecting Ho, given that Ha is true. =1- β , β is the Type II error Function of n, α σ effect size of interest(the “distance” between Ho and Ha)
20 June 2003 Biostatistics 6650--L14 5 X is the variable of interest, reject Ho if sample mean is too big, > C 0 3 E 3 E 3 E ) E Power: 1-sided test: Ho: 0=0 o , Ha: 0=0 1 >0 o Distn of under: X X X 0 0 0 = Type I error, usually .05 = Pr(reject Ho | Ho true) = Pr( > C | 0 0 ) 0 ) 0 = Type II error = Pr(FTR Ho | Ha true) = Pr( < C | 0 0 ) Power 0 0 C 0 Ho Power=1-0=Pr(reject Ho | Ha) depends on: n, α , 0 1 -0 o , 0 2 . often want power=.8 or .9 0 0 0 Ha

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6 Biostatistics 6650--L14 20 June 2003 1-sample: Power 0 obs 1-α o o a a 0 1-α 0 1-α 0 1-α o 1 1 1 1 x-μ reject if z = / Power 1β Pr(Reject H | H is true) =Pr(X C | H ), C=μ + z σ/ n n Pr(Xμ + z σ/ n | ) = Pr(X-μ - + z σ/ One sample 1-sided test: Ho: μ=μ Ha: μ=μ >μ μ=μ μ μ z σ = - = = 0 1-α 1 o a α 1 0 α 1 1 1 o 1 n ) X-μ - = Pr( + z ) σ/ n σ/ n X- μ - = Pr( + z ), Note left part is N(0,1) under H . σ/ n σ/ n (μ -μ ) =Φ z + , σ μ μ μ μ μ >μ n
20 June 2003 Biostatistics 6650--L14 7 1-sample: Power Power= Φ [z], = Pr(Z< z ) power increases as “z” increases, or equivalently as: » n increases » μ 1 - μ o ” (or distance from H o to H a ) increases » as 0 decreases » as Z α ( or 0) increases,… reject more often o 1 0 α o 1 o 1 (μ -μ ) Power = Φ z + , σ One sample 1-sided test: Ho:μ=μ Ha: μ=μ >μ μ >μ n

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20 June 2003 Biostatistics 6650--L14 8 1-sample: Power 0 obsα o a a 0 1 α o o 1 1 x-μ reject if
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Lecture14 - Todays Schedule Sample size and power for...

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